Induction of mathematical knowledge points in the last semester of senior two.
fractional equation
I. Understanding the definition
1. Fractional equation: an equation with a fraction and an unknown number in the denominator-fractional equation.
2, the idea of solving the fractional equation is:
(1) Multiplies the simplest common denominator on both sides of the equation, removes the denominator, and becomes an integral equation.
(2) Solve the whole equation.
(3) Bring the root of the whole equation into the simplest common denominator to see if the result is zero, so that the root of the simplest common denominator is the additional root of the original equation and must be discarded.
(4) Write the root of the original equation.
"Four summaries of one transformation, two solutions and three experiments"
3. Root addition: The root addition of fractional equation must meet two conditions:
(1) Finding the root is the simplest, and the common denominator is 0; (2) Increasing root is the root of integral equation formed by fractional equation.
4, the solution of fractional equation:
(1) Simplification before simplification (2) Multiply both sides of the equation by the simplest common denominator and turn it into an integral equation;
(3) solving the integral equation; (4) Root inspection;
Note: When solving the fractional equation, when both sides of the equation are multiplied by the simplest common denominator, the simplest common denominator may be 0, which increases the root, so the fractional equation must be tested.
Test method of fractional equation: bring the solution of the whole equation into the simplest common denominator. If the value of the simplest common denominator is not 0, the solution of the whole equation is the solution of the original fractional equation; Otherwise, this solution is not the solution of the original fractional equation.
5. Fractional equation solves practical problems.
Steps: Examining questions-setting unknowns-listing equations-solving equations-testing-writing answers. Pay attention to the test equation itself and practical problems when testing.
Second, the axisymmetric graphics:
A figure is folded in half along a straight line, and the parts on both sides of the straight line can completely overlap. This straight line is called the axis of symmetry. Points that coincide with each other are called corresponding points.
1, axisymmetric:
Two figures are folded in half along a straight line, and one of them can completely coincide with the other. This straight line is called the axis of symmetry. Points that coincide with each other are called corresponding points.
2, the difference and connection between axisymmetric graphics and axisymmetric:
(1) difference. Axisymmetric graphics discuss "the symmetrical relationship between graphics and straight lines"; Axisymmetry discusses "the symmetrical relationship between two figures and a straight line".
(2) contact. Axisymmetric figures are defined as "the parts on both sides of the axis of symmetry are regarded as two figures". Axisymmetric "two figures as a whole" is an axisymmetric figure.
3, the essence of axial symmetry:
(1) Two symmetric graphs are congruent.
(2) The symmetry axis is perpendicular to the line segment connecting the corresponding points.
(3) The distances from the corresponding points to the symmetry axis are equal.
(4) The connecting lines of the corresponding points are parallel to each other.
Third, use coordinates to represent the axis symmetry.
1, and the coordinates of the point (x, y) which is symmetrical about x axis are (x,-y);
2. The coordinates of the point (x, y) about the Y axis symmetry are (-x, y);
3. The coordinates of the point (x, y) symmetrical about the origin are (-x, -y).
4. About the symmetry of the bisector of the coordinate axis.
The point P(x, y) is symmetrical about the bisector y=x of the first and third quadrant coordinate axes, and the coordinate of this point is (y, x).
The point P(x, y) is symmetrical about the bisector y=-x of the second and fourth quadrant coordinate axes, and the coordinate of this point is (-y, -x).
Eighth grade mathematics knowledge point book 1
First of all, in a plane, two data are usually needed to determine the position of an object.
Second, the plane rectangular coordinate system and related concepts
1, plane rectangular coordinate system
In a plane, two mutually perpendicular axes with a common origin form a plane rectangular coordinate system. Among them, the horizontal axis is called X axis or horizontal axis, and the right direction is the positive direction; The vertical axis is called Y axis or vertical axis, and the orientation is positive; The x-axis and y-axis are collectively referred to as coordinate axes. Their common origin o is called the origin of rectangular coordinate system; The plane on which the rectangular coordinate system is established is called the coordinate plane.
2. In order to describe the position of a point in the coordinate plane conveniently, the coordinate plane is divided into four parts, namely the first quadrant, the second quadrant, the third quadrant and the fourth quadrant.
Note: The points on the X axis and Y axis (points on the coordinate axis) do not belong to any quadrant.
3. The concept of point coordinates
For any point P on the plane, the intersection point P is perpendicular to the X-axis and Y-axis respectively, and the numbers A and B corresponding to the vertical feet on the X-axis and Y-axis are respectively called the abscissa and ordinate of the point P, and the ordered number pair (A, B) is called the coordinate of the point P. ..
The coordinates of points are represented by (a, b), and the order is abscissa before, ordinate after, and there is a ","in the middle. The positions of horizontal and vertical coordinates cannot be reversed. The coordinates of points on the plane are ordered real number pairs. At that time, (a, b) and (b, a) were the coordinates of two different points.
There is a one-to-one correspondence between points on the plane and ordered real number pairs.
4. Coordinate characteristics of different locations
(1), the coordinate characteristics of the midpoint of each quadrant.
Point P(x, y) is in the first quadrant: x; 0，y； 0
Point P(x, y) is in the second quadrant: x; 0，y； 0
Point P(x, y) is in the third quadrant: x; 0，y； 0
Point P(x, y) is in the fourth quadrant: x; 0，y； 0
(2) Characteristics of points on the coordinate axis
The point P(x, y) is on the x axis, y=0, and x is an arbitrary real number.
The point P(x, y) is on the y axis, x=0, and y is an arbitrary real number.
Point P(x, y) is on both X and Y axes, and both X and Y are zero, that is, the coordinate of point P is (0,0), that is, the origin.
(3) Coordinate characteristics of points on the bisector of two coordinate axes.
Point P(x, y) is on the bisector of the first and third quadrants (straight line y=x), and x and y are equal.
Point P(x, y) is on the bisector of the second and fourth quadrants, and x and y are reciprocal.
(4) Characteristics of the coordinates of points on a straight line parallel to the coordinate axis
The ordinate of each point on the straight line parallel to the X axis is the same.
The abscissa of each point on the straight line parallel to the Y axis is the same.
(5) Coordinate characteristics of points symmetrical about the X axis, Y axis or origin.
The abscissa of point P and point P' is equal to the X axis, and the ordinate is opposite, that is, the symmetrical point of point P(x, y) relative to the X axis is P'(x, -y).
The axisymmetrical ordinate of point P and point P' with respect to Y is equal, and the abscissa is opposite, that is, the symmetrical point of point P(x, y) with respect to Y axis is P'(-x, y).
Point P and point P' are symmetrical about the origin, and the abscissa and ordinate are opposite, that is, the symmetrical point of point P(x, y) about the origin is P'(-x, -y).
Math review method in junior two
orderly
Mathematics is an interlocking subject, and any link will affect the whole learning process. Therefore, don't be greedy when studying. You should pass the exam chapter by chapter, and don't leave questions that you don't understand or understand deeply easily.
Emphasize understanding
Concepts, theorems and formulas should be memorized on the basis of understanding. Every time you learn a new theorem, first try to do an example without looking at the answer to see if you can correctly apply the new theorem; If not, compare the answers to deepen the understanding of the theorem.
basic skill
You can't learn mathematics without training. Usually do more exercises with moderate difficulty. Of course, don't fall into the misunderstanding of dead drilling questions, be familiar with the questions of the college entrance examination, and be targeted in training.
Pay attention to mistakes
Booking the wrong book and collecting the wrong questions by yourself is often your weakness. When reviewing, this wrong book has become a valuable review material.
Mathematics learning is a step-by-step process, and it is unrealistic to dream of reaching the sky in one step. After reciting the contents of the book, carefully write the exercises at the back of the book. Some students may think that the exercises after the book are too simple to do. This idea is highly undesirable. The function of the exercises after the book is not only to help you remember the contents in the book, but also to help you standardize the writing format, make your problem-solving structure compact and tidy, make proper use of formulas and theorems, and reduce unnecessary marks in the exam.
The usual mathematical research:
○ 1 preview carefully before class. The purpose of preview is to listen to the teacher better. Through preview, the mastery level should reach 80%. Listen to the teacher answer these questions with questions that you don't understand in the preview. Preview can also improve the overall efficiency of attending classes. Specific preview method: finish the topics in the book and draw the knowledge points. The whole process lasts about 10.
○2 Let math class combine with practice. It's no use just listening in math class. When the teacher asks the students to do calculus on the blackboard, they should also practice on the draft paper. You must ask questions you don't understand, or you may not do it if you encounter similar problems in the exam. When listening to the teacher's lecture, you must concentrate on the details, otherwise, "the embankment of a thousand miles will collapse in the ant nest."
○3 Review in time after class. After finishing your homework, sort out what the teacher said that day, and you can do extracurricular problems for about 25 minutes. You can choose the extracurricular books that suit you according to your own needs. The content of the extracurricular problem is probably today's class.
The fourth unit test is to test your recent study. In fact, the score represents your past. The key is to sum up and learn from each exam so that you can do better in the mid-term and final exams. Teachers often take exams without notice and review them in time after class.
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